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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Write (<a href="" class="xref" data-knowl="./knowl/eq5_11.html" title="Equation 5.5.1">(5.5.1)</a>) as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_11.html ./knowl/eq5_12.html ./knowl/eq5_13.html ./knowl/eq5_14.html ./knowl/eq5_15.html ./knowl/eq5_17.html">
\begin{equation}
x^2 y^{\prime \prime}+ x \left( x \frac{Q(x)}{P(x)}\right)y^{\prime}+\left(x^2 \frac{R(x)}{P(x)} \right) y=0.\tag{5.5.4}
\end{equation}
</div>
<p class="continuation">Substituting (<a href="" class="xref" data-knowl="./knowl/eq5_12.html" title="Equation 5.5.2">(5.5.2)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq5_13.html" title="Equation 5.5.3">(5.5.3)</a>) into the (<a href="" class="xref" data-knowl="./knowl/eq5_14.html" title="Equation 5.5.4">(5.5.4)</a>):</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_11.html ./knowl/eq5_12.html ./knowl/eq5_13.html ./knowl/eq5_14.html ./knowl/eq5_15.html ./knowl/eq5_17.html">
\begin{equation*}
\begin{aligned}
&amp;x^2 \left[r(r-1) a_0 x^{r-2}+r (r+1) a_1 x^{r-1}+\cdots \right]+x (p_0+p_1 x+\cdots) \left[r a_0 x^{r-1}+(r+1) a_1 x^r+\cdots \right]\\
&amp;\quad +(q_0+q_1 x+\cdots)\left[ a_0 x^r+a_1 x^{1+r}+\cdots \right]=0\\
&amp; \to r(r-1) a_0 x^r+(r+1) r a_1 x^{r+1}+\cdots\\
&amp;\qquad +r p_0 a_0 x^r+[r p_1 a_0+(r+1) p_0 a_1]x^{r+1}+\cdots\\
&amp;\qquad +q_0 a_0 x^r+(q_1 a_0+q_0 a_1) x^{r+1}+\cdots=0
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Dividing both sides by the factor <span class="process-math">\(x^r\text{:}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_11.html ./knowl/eq5_12.html ./knowl/eq5_13.html ./knowl/eq5_14.html ./knowl/eq5_15.html ./knowl/eq5_17.html">
\begin{equation*}
a_0+[(r+1) r a_1+r p_1 a_0+(r+1) p_0 a_1+q_1 a_0+q_0 a_1] x+\cdots=0,
\end{equation*}
</div>
<p class="continuation">which further gives</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_11.html ./knowl/eq5_12.html ./knowl/eq5_13.html ./knowl/eq5_14.html ./knowl/eq5_15.html ./knowl/eq5_17.html">
\begin{equation}
[r^2+r( p_0-1)+q_0] a_0=0,\tag{5.5.5}
\end{equation}
</div>
<p class="continuation">and</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_11.html ./knowl/eq5_12.html ./knowl/eq5_13.html ./knowl/eq5_14.html ./knowl/eq5_15.html ./knowl/eq5_17.html">
\begin{equation}
(r+1) r a_1+r p_1 a_0+(r+1) p_0 a_1+q_1 a_0+q_0 a_1=0.\tag{5.5.6}
\end{equation}
</div>
<p class="continuation">From (<a href="" class="xref" data-knowl="./knowl/eq5_15.html" title="Equation 5.5.5">(5.5.5)</a>), since <span class="process-math">\(a_0 \neq 0\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_11.html ./knowl/eq5_12.html ./knowl/eq5_13.html ./knowl/eq5_14.html ./knowl/eq5_15.html ./knowl/eq5_17.html">
\begin{equation}
r^2+r( p_0-1)+q_0=0.\tag{5.5.7}
\end{equation}
</div>
<p class="continuation">Equation (<a href="" class="xref" data-knowl="./knowl/eq5_17.html" title="Equation 5.5.7">(5.5.7)</a>) is the <dfn class="terminology">indicial equation</dfn> and the <span class="process-math">\(r\)</span> is called the <dfn class="terminology">exponents at the singularity</dfn>.</p>
<span class="incontext"><a href="sec5_5.html#p-231" class="internal">in-context</a></span>
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